Boolean Input

Definition

A Boolean input is an input with two possible values.

Common values:

0
1

or:

false
true

Boolean input = input with two possible states.

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Boolean Values As Symbols

Boolean values can be represented by symbols.

Example:

0 = false
1 = true

The symbols 0 and 1 are not the only possible representation.

We could also write:

F and T
No and Yes
Off and On

The important part is that the two states are distinguishable and used consistently.

Boolean values are two-state symbols.

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Boolean Input As Vector Data

Two Boolean inputs can be represented as a vector.

Example:

x = [x1, x2]

where each component can be:

0 or 1

Possible inputs:

[0, 0]
[0, 1]
[1, 0]
[1, 1]

Boolean inputs can still be treated as vector data.

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Discrete Input

Boolean input is discrete.

There are no values between:

0 and 1

in the Boolean setting.

The input is either one state or the other.

Boolean values are separate possibilities, not continuous measurements.

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Why Boolean Inputs Matter For XOR

XOR uses two Boolean inputs.

Example:

x1 = 0
x2 = 1

The output is also Boolean:

0 or 1

This makes XOR small enough to list completely.

XOR is built from two Boolean inputs and one Boolean output.

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Boolean Does Not Mean Simple For Linear Models

Boolean inputs are simple as values.

But their class pattern can still be difficult for a linear classifier.

XOR is the key example.

It has only four possible inputs, but one straight line cannot separate its classes.

Simple inputs can still create nonlinearly separable patterns.

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Final Summary

• A Boolean input has two possible values.
• Common encodings are 0/1 or false/true.
• Boolean values are symbols used consistently.
• Two Boolean inputs can form a vector.
• Boolean inputs are discrete.
• XOR uses two Boolean inputs.
• Simple Boolean input can still expose limits of linear classification.